We first extend the results of Chatterjee,Kumar,Shi,Volk(Computational Complexity 2022) by showing that the degree $d$ elementary symmetric polynomials in $n$ variables have formula lower bounds of $Ω(d(n-d))$ over fields of positive characteristic. Then, we show that the results of the universality of linear projections of elementary symmetric polynomials from Shpilka(JCSS 2002) and of border fan-in two $ΣΠΣ$ circuits from Kumar(ACM TOCT 2020) over zero characteristic fields do not extend to fields of positive characteristic. In particular, we show that *There are polynomials that cannot be represented as linear projections of the elementary symmetric polynomials(in fact, we show linear lower bounds over the size of the sum of such linear projections) and *There are polynomials that cannot be computed by border depth-$3$ circuits of top fan-in $k$, called $\overline{Σ^{[k]}ΠΣ}$, for $k = o(n)$. To prove the first result, we consider a geometric property of the elementary symmetric polynomials, namely, the set of all points in which the polynomial and all of its first-order partial derivatives vanish. It was previously shown that the dimension of this space was exactly $d-2$ for fields of zero characteristic. We extend this to fields of positive characteristic by showing that this dimension must be between $d-2$ and $d-1$. In fact, we provide some criterion where it is $d-2$ and others where it is $d-1$. Then, to consider the border top fan-in of linear projections of the elementary symmetric polynomials and border depth-$3$ circuits(sometimes called border affine Chow rank), we show that it is sufficient to consider the border top fan-in of the sum of such linear projections of the elementary symmetric polynomials. This is done by an explicit construction of a 'metapolynomial,' meaning that this result also applies in the border setting.

翻译：我们首先推广了Chatterjee、Kumar、Shi、Volk（Computational Complexity 2022）的结果，证明了在正特征域上，$n$个变量的$d$次初等对称多项式具有$Ω(d(n-d))$的公式下界。接着，我们证明了Shpilka（JCSS 2002）关于初等对称多项式线性投影的普适性结果，以及Kumar（ACM TOCT 2020）关于零特征域上边界扇入为二的$ΣΠΣ$电路的结果，均无法推广到正特征域。具体而言，我们证明了：*存在无法表示为初等对称多项式线性投影的多项式（事实上，我们给出了此类线性投影之和的规模的线性下界）；*对于$k = o(n)$，存在无法由顶层扇入为$k$的边界深度-$3$电路（称为$\overline{Σ^{[k]}ΠΣ}$）计算的多项式。为证明第一个结果，我们考察了初等对称多项式的一个几何性质，即多项式及其所有一阶偏导数均消失的点集。先前研究表明，在零特征域上该空间的维数恰好为$d-2$。我们将其推广到正特征域，证明该维数必须在$d-2$到$d-1$之间。事实上，我们给出了该维数为$d-2$的若干判据，以及为$d-1$的其他判据。随后，为研究初等对称多项式线性投影的边界顶层扇入与边界深度-$3$电路（有时称为边界仿射Chow秩），我们证明只需考虑此类初等对称多项式线性投影之和的边界顶层扇入。这是通过显式构造一个“元多项式”实现的，意味着该结果在边界设定下同样成立。